Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#59581 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2428 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6428 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 26 points:
(1,2);(3,4,7,5)(6,9,10,8)(12,13)(16,17)(19,20,22,23)(21,25,26,24);(4,5)(8,9)(12,13)(16,17)(20,23)(24,25);(11,12,15,13)(14,16,18,17)(19,20)(21,24)(22,23)(25,26);(3,5,7,4)(6,9,10,8)(11,13,15,12)(14,17,18,16)(19,20)(21,24)(22,23)(25,26);(3,6)(4,8)(5,9)(7,10)(11,14,15,18)(12,16,13,17)(19,21,22,26)(20,24,23,25)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573592);